Integrand size = 33, antiderivative size = 256 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 a \left (3 A b^2+8 a^2 C-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (3 A b^2+\left (8 a^2+b^2\right ) C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d} \]
2*a*(A*b^2+C*a^2)*sin(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+2/3*C* sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d-2/3*a*(3*A*b^2+8*C*a^2-5*C*b^2)*(c os(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c) ,2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b^3/(a^2-b^2)/d/((a+b*cos (d*x+c))/(a+b))^(1/2)+2/3*(3*A*b^2+(8*a^2+b^2)*C)*(cos(1/2*d*x+1/2*c)^2)^( 1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/ 2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*cos(d*x+c))^(1/2)
Time = 2.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.82 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 \left (a (a+b) \left (3 A b^2+8 a^2 C-5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-\left (a^2-b^2\right ) \left (3 A b^2+\left (8 a^2+b^2\right ) C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (-4 a^3 C+a b^2 (-3 A+C)+b \left (-a^2+b^2\right ) C \cos (c+d x)\right ) \sin (c+d x)\right )}{3 (a-b) b^3 (a+b) d \sqrt {a+b \cos (c+d x)}} \]
(-2*(a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a^2 - b^2)*(3*A*b^2 + (8*a^ 2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b )/(a + b)] + b*(-4*a^3*C + a*b^2*(-3*A + C) + b*(-a^2 + b^2)*C*Cos[c + d*x ])*Sin[c + d*x]))/(3*(a - b)*b^3*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.41 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3511, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3511 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int \frac {-b \left (a^2-b^2\right ) C \cos ^2(c+d x)+a \left (2 C a^2+A b^2-b^2 C\right ) \cos (c+d x)+b \left (C a^2+A b^2\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-b \left (a^2-b^2\right ) C \cos ^2(c+d x)+a \left (2 C a^2+A b^2-b^2 C\right ) \cos (c+d x)+b \left (C a^2+A b^2\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-b \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (2 C a^2+A b^2-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left (C a^2+A b^2\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \int \frac {\left (3 A b^2+\left (2 a^2+b^2\right ) C\right ) b^2+a \left (8 C a^2+3 A b^2-5 b^2 C\right ) \cos (c+d x) b}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\int \frac {\left (3 A b^2+\left (2 a^2+b^2\right ) C\right ) b^2+a \left (8 C a^2+3 A b^2-5 b^2 C\right ) \cos (c+d x) b}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\int \frac {\left (3 A b^2+\left (2 a^2+b^2\right ) C\right ) b^2+a \left (8 C a^2+3 A b^2-5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx-\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {2 a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {2 a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {2 a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2 a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {2 a \left (8 a^2 C+3 A b^2-5 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (8 a^2 C+3 A b^2+b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 C \left (a^2-b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{b^2 \left (a^2-b^2\right )}\) |
(2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d *x]]) - (((2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Elli pticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(3*A*b^2 + 8*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])) /(3*b) - (2*(a^2 - b^2)*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/(b ^2*(a^2 - b^2))
3.7.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[ (-(b*c - a*d))*(A*b^2 + a^2*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/( b^2*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d )) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] + b *C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e , f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(888\) vs. \(2(296)=592\).
Time = 16.50 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.47
method | result | size |
default | \(\text {Expression too large to display}\) | \(889\) |
parts | \(\text {Expression too large to display}\) | \(1118\) |
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(8/b*C*(-1/ 6/b*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c) ^2)^(1/2)+1/6*(a-b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c )^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^ 2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/12/b^2*(-2*a+6 *b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a- b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(El lipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c ),(-2*b/(a-b))^(1/2))))+2*C/b^3*(a+2*b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2) *((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+( a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b) )^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))+2*(A*b^2+C*a^2+ C*a*b+C*b^2)/b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a -b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1 /2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+2*a*(A*b^2+C*a^2)/b^3 /sin(1/2*d*x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^2)*(-2*sin(1/2 *d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*si n(1/2*d*x+1/2*c)^2*b+EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(-2* b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*a-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(-2*b/(a-b)*sin(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.82 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {6 \, {\left (4 \, C a^{3} b^{2} + {\left (3 \, A - C\right )} a b^{4} + {\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) - {\left (\sqrt {2} {\left (16 i \, C a^{4} b + 2 i \, {\left (3 \, A - 8 \, C\right )} a^{2} b^{3} - 3 i \, {\left (3 \, A + C\right )} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, C a^{5} + 2 i \, {\left (3 \, A - 8 \, C\right )} a^{3} b^{2} - 3 i \, {\left (3 \, A + C\right )} a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - {\left (\sqrt {2} {\left (-16 i \, C a^{4} b - 2 i \, {\left (3 \, A - 8 \, C\right )} a^{2} b^{3} + 3 i \, {\left (3 \, A + C\right )} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, C a^{5} - 2 i \, {\left (3 \, A - 8 \, C\right )} a^{3} b^{2} + 3 i \, {\left (3 \, A + C\right )} a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 3 \, {\left (\sqrt {2} {\left (-8 i \, C a^{3} b^{2} - i \, {\left (3 \, A - 5 \, C\right )} a b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, C a^{4} b - i \, {\left (3 \, A - 5 \, C\right )} a^{2} b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (8 i \, C a^{3} b^{2} + i \, {\left (3 \, A - 5 \, C\right )} a b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, C a^{4} b + i \, {\left (3 \, A - 5 \, C\right )} a^{2} b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right )}{9 \, {\left ({\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}} \]
1/9*(6*(4*C*a^3*b^2 + (3*A - C)*a*b^4 + (C*a^2*b^3 - C*b^5)*cos(d*x + c))* sqrt(b*cos(d*x + c) + a)*sin(d*x + c) - (sqrt(2)*(16*I*C*a^4*b + 2*I*(3*A - 8*C)*a^2*b^3 - 3*I*(3*A + C)*b^5)*cos(d*x + c) + sqrt(2)*(16*I*C*a^5 + 2 *I*(3*A - 8*C)*a^3*b^2 - 3*I*(3*A + C)*a*b^4))*sqrt(b)*weierstrassPInverse (4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) - (sqrt(2)*(-16*I*C*a^4*b - 2*I*(3*A - 8 *C)*a^2*b^3 + 3*I*(3*A + C)*b^5)*cos(d*x + c) + sqrt(2)*(-16*I*C*a^5 - 2*I *(3*A - 8*C)*a^3*b^2 + 3*I*(3*A + C)*a*b^4))*sqrt(b)*weierstrassPInverse(4 /3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*(sqrt(2)*(-8*I*C*a^3*b^2 - I*(3*A - 5* C)*a*b^4)*cos(d*x + c) + sqrt(2)*(-8*I*C*a^4*b - I*(3*A - 5*C)*a^2*b^3))*s qrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^ 3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^ 3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*(sqrt(2)*(8*I *C*a^3*b^2 + I*(3*A - 5*C)*a*b^4)*cos(d*x + c) + sqrt(2)*(8*I*C*a^4*b + I* (3*A - 5*C)*a^2*b^3))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/ 27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/ 27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a )/b)))/((a^2*b^5 - b^7)*d*cos(d*x + c) + (a^3*b^4 - a*b^6)*d)
Timed out. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]